Abelianisoitu

Abelianisoitu, in mathematical context, refers to the process or result of turning a group into an abelian group by factoring out its non-commuting part. In group theory, the abelianization of a group G is the quotient G/[G,G], where [G,G] is the commutator subgroup generated by all commutators [g,h] = g^{-1}h^{-1}gh. The abelianization is the largest abelian quotient of G: any homomorphism from G to an abelian group A factors uniquely through π: G -> G/[G,G].

Concretely, to obtain G^{ab}, one imposes the relations gh = hg for all g,h in G, effectively collapsing all commutators to the identity. If G is already abelian, [G,G] = {e} and G^{ab} ≅ G. If G is a free group on a set X, its abelianization is the free abelian group on X. For finite groups, the order of the abelianization equals the index [G:[G,G]]. The abelianization is naturally isomorphic to the first homology group H_1(G) with integer coefficients.

Properties and scope: The construction is functorial, yielding a functor (-)^{ab} from groups to abelian groups, which is left adjoint to the inclusion of abelian groups into groups. It provides a standard measure of a group’s non-abelian nature: two non-abelian groups can have isomorphic abelianizations, so the process preserves only the abelian quotient information.

Applications: Abelanization appears in topology via H_1 and in algebra as a tool to study maps to abelian targets, using its universal property to factor such maps uniquely through the abelianization.